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my question is inspired by projects like Perlan etc when one is trying to fly in thin air with a glider. It is well known that at high altitude (say 10-20km) aircraft control becomes 'sluggish'. However, thinking about lift equation I have encountered a paradox.

According to https://en.wikipedia.org/wiki/Lift_(force)

Lift coefficient is L=const x rho_air x true_airspeed^2 x surface_area x coefficient_of_lift

Let's neglect the fact that at high altitude, with Reynolds number changing, one will get into a diferent CL (coefficient_of_lift) vs AOA curve. This is a small issue compared to span of possible rho_air values. Because this equation has true_airspeed^2, it is obvious that at low altitude, aileron effectiveness (which is also a lifting surface) will raise with second power when increasing TAS which at normal atmospheric conditions and low altitude, low subsonic speeds equals to EAS and CAS (IAS). If we take any glider and lift it up, rho_air will decrease even by factor of 16 for simplicity. This will be compensated by required TAS forward glide speed by factor of 4, so that wings can still provide equal lifting force, otherwise tha plane would fall because of winning F=m*g - which doesn't care about air composition. The plane will fly faster, and while we should expect that diminished rho_air would diminish aileron effectiveness, increased TAS will naturally compensate for this following the same equation...

I have concluded that starting to roll which requires applying non-opposed rolling force to overcome (constant!) airplane inertia, so that roll rate will rise until it becomes balanced by wing drag resistance, should require identical aileron deflection for airplane at low and high altitude. In other words achievable rates of change of roll rate should be the same at cruise speed in thin air when using the same aileron deflection.

At high altitude however, measured IAS and CAS will be much lower than TAS but that doesn't matters as stable airplane will balance by itself at cruise speed.

Then we come into question what is the maximum roll rate achieved in thin vs dense air. After applying aileron deflection the roll rate will increase, when applied rolling force is compeensated by aircraft drag due to rollign motion whcih is again proportional to constrho_airrollrate^2.

I have concluded that maximum achievable rollrate will be higher at thin air, using the same deflection of ailerons that provide the same overall rollign force at high altitude because the airplame must fly faster just to provide its lift. With the same aileron deflection, achieved stable state roll rates will be higher in thin air than at dense air with the same aileron deflection

If all above is true, why airplanes are said to be 'sluggish at high altitudes due to thin air'.

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Your conclusions are correct; however, it is new to me that aircraft control becomes "sluggish" at high altitude. Only when you fly up into the coffin corner will control effectivity suffer since you fly close to stall, but that is very similar to the diminishing control effectiveness when you approach stall at lower altitude.

To express your conclusions in the way it can be found in flight mechanics books:

  1. Roll acceleration does not change with altitude because it is dominated by aileron control power and roll inertia, and
  2. The final roll rate will change inversely with the square root of air density because here the roll damping of the wing comes in which is inversely proportional to airspeed.
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What really matters is the air dynamic pressure for the computation of the aileron effectiveness, so that you can get rid of density (altitude) effects. That's where KEAS meaning takes place. As far as I know, all aeroelastic studies for aileron roll effectiveness are performed with KEAS, not KTAS.

Please refer to aeroelastic and flight testing textbooks for a better and comprehensive explanation, such as:

  • Dowell - A Modern Course in Aeroelasticity 4th Ed
  • Wright & Cooper - Introduction to Aircraft Aeroelasticity and Loads
  • Ward & Strganac - Introduction to Flight Test Engineering 2nd Ed
  • Kimberlin - Flight Testing of Fixed Wing Aircraft

You will see that the aileron effectiveness depends only on the ratio between the current dynamic pressure to the reversal dynamic pressure.

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